Nonlinear Tikhonov regularization in Hilbert scales with oversmoothing penalty: inspecting balancing principles

TL;DR

This paper analyzes nonlinear Tikhonov regularization with oversmoothing penalties in Hilbert scales, focusing on how to select the regularization parameter via balancing principles to achieve order-optimal reconstructions without strong a priori smoothness assumptions. It derives an error decomposition under general source conditions and introduces an auxiliary proximal element to enable analysis in the oversmoothing regime. Three quasi-optimal balancing variants are proposed and shown to achieve near-oracle performance under a unified framework, with explicit attention to the degree of ill-posedness and the smoothness class (including Hölder and logarithmic). The results are supported by a numerical study on an exponential growth model, illustrating the practical behavior of the balancing rules and the impact of oversmoothing on parameter choice, convergence rates, and reconstruction quality.

Abstract

The analysis of Tikhonov regularization for nonlinear ill-posed equations with smoothness promoting penalties is an important topic in inverse problem theory. With focus on Hilbert scale models, the case of oversmoothing penalties, i.e., when the penalty takes an infinite value at the true solution gained increasing interest. The considered nonlinearity structure is as in the study B. Hofmann and P. Mathé. Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales. Inverse Problems, 2018. Such analysis can address two fundamental questions. When is it possible to achieve order optimal reconstruction? How to select the regularization parameter? The present study complements previous ones by two main facets. First, an error decomposition into a smoothness dependent and a (smoothness independent) noise propagation term is derived, covering a large range of smoothness conditions. Secondly, parameter selection by balancing principles is presented. A detailed discussion, covering some history and variations of the parameter choice by balancing shows under which conditions such balancing principles yield order optimal reconstruction. A numerical case study, based on some exponential growth model, provides additional insights.

Figures (3)

• Figure 1: Exponential growth model with $x^\dag(t)\equiv 1; \; (0 < t \leq 1)$ and parameter choice using the balancing principle \ref{['eq:Hdelta-def']} with $C_{BP}=0.1$. $\alpha_{ BP}$ in red for various $\delta$ and best approximating regression line in blue/dashed on a log-log scale (left) and approximation error $\|x_{\alpha}^\delta-x^\dag\|_X$ in red and approximate rate in blue/dashed (right).
• Figure 2: Exponential growth model with $x^\dag(t)\equiv 1; \; (0 < t \leq 1)$ and $\delta=0.0179$. Visualization of $\| x_{\alpha_{k+1}}^{\delta} - x_{\alpha_{k}}^{\delta} \|_X$ and $\| x_{\alpha}^\delta-x^\dag\|_X$ as well as parameter choice using the balancing principle, discrepancy principle, quasi-optimality and $\alpha_{ \textup{opt}}$ for various $C_{BP}$ and $C_{ DP}$.
• Figure 3: Exponential growth model with $x^\dag(t)\equiv 1; \; (0 < t \leq 1)$ (left) and $\hat{x}^{\dag}(t)=-(t-\frac{1}{2})^2+\frac{1}{4}; \; (0 < t \leq 1)$, $\delta=0.0179$. Regularized and exact solutions for various regularization parameters.

Theorems & Definitions (33)

• Definition 1
• Theorem 1
• proof
• Example 1: power-type smoothness
• Example 2: low order smoothness
• Example 3: no explicit smoothness
• Definition 2
• Proposition 1
• proof
• Lemma 1